The integral is to be computed by Romberg integration

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

The integral

\int_0^1x^{3/2}\,\mathrm{d}x = 0.4

is to be computed by Romberg integration. In the notation of Section 5.4, with

h = 1

, the trapezium sums with

4

8

and

16

intervals are

R_{2,0} = 0.40701811

R_{3,0} = 0.40181246

and

R_{4,0} = 0.40046340

. Errors are measured as value minus exact value. ((a)) Use Eq. (5.15) to compute

R_{3,1}

and

R_{4,1}

. Give their errors, and compare how the errors of the first column

R_{k,0}

and of the second column

R_{k,1}

change when the interval is halved. ((b)) Find the power

p

such that the error of

R_{k,1}

is proportional to

h_k^p

, and say why the column

R_{k,2}

cannot remove it. ((c)) Compute

R_{4,2}

and its error, and the factor by which it improves on

R_{4,1}

. ((d)) The substitution

x = t^2

turns the integral into

\int_0^12t^4\,\mathrm{d}t

. Compute its trapezium sums with

1

2

and

4

intervals, build

R_{2,2}

, and explain the result.

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